# nLab bifibration

Contents

This page is about Grothendieck fibrations that are also opfibrations. Not to be confused with two-sided fibrations nor with fibrations of 2-categories (both of which some authors also refer to as “bifibrations”).

category theory

# Contents

## Definition

A bifibration of categories is a functor

$\array{ E \\ \big\downarrow \\ B }$

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism $f \colon b_1 \to b_2$ in a bifibration has both a push-forward $f_* : E_{b_1} \to E_{b_2}$ as well as a pullback $f^* : E_{b_2} \to E_{b_1}$.

## Properties

### Relation to pseudofunctors in adjoints

###### Proposition

Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{Adj}$ are equivalently the bifibrations.

###### Proof

A Grothendieck construction on a pseudofunctor yielding a bifibration is fairly immediately equivalent to all base change functors having an adjoint on the respective side (e.g. Jacobs (1998), Lem. 9.1.2). By the fact that $Cat_{adj} \to Cat$ is a locally full sub-2-category (this Prop.) this already means that the given pseudofunctor factors through $Cat_adj$, and essentially uniquely so.

###### Remark

Further factoring through ModCat $\longrightarrow Cat_{Ajd}$ hence yields bifibrations of model categories [Harpaz & Prasma (2015), Sec. 3; Cagne & Melliès (2020)]. See at model structures on Grothendieck constructions for more on this.

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.

## Bifibration of bicategories

The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into $2 Cat_{adj}$, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor $B^{coop} \to 2Cat$) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor $B\to 2Cat$) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.

Original notion and terminology of “bifibration”:

• Alexander Grothendieck, Catégories co-fibrées, catégories bi-fibrées., Section 10 in exposé VI of: Revêtements Etales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61 (SGA 1), LNM 224 Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]

Further early discussion (not using the terminology “bifibration”, though):

Discussion in the context of the Beck-Chevalley condition:

In the context of categorical semantics for dependent types: